﻿376 Mr. J. J. Sylvester on two remarkable Resultants arising 



Suppose any number of equations (to fix the ideas say four) 

 of the form which follows : 



J] = ax + by + cz -\-dt = 0, 



Y=ax 3 + by 3 + cz 3 + dt 3 =0, 



W — ax b -\-by h + cz b + dl 5 = 0, 



£l = ax 7 + by 7 + cz 7 + dt 7 = 0. 

 If these be regarded as surfaces, they can only be made to inter- 

 sect in one or another of a definite number of points. 

 Eor in the case of intersection we must evidently have 



1111 



dt 



X* 



y* 



«2 



f 



X 4 



2/ 4 



~4 



Z 



t 4 



X 6 



y 6 



z 6 



I 6 



i.e. dtfa*. ft *, t^=0, 



£ being the symbol which expresses the product of the differences 

 of the quantities which it affects. Hence 



x±y = or x±z = or y±_z — Q or x + t = or y±t = 

 or z±t=0 or / = 0. 

 Hence it will easily be seen by substitution and successive reduc- 

 tion that the points of intersection are confined to those herein- 

 under stated and their analogues, viz. 



a?=+y=+z =+t, 

 x= ±y= +z, t=0j 



x=±y, z = 0, t = 0, 



x=0, y = Q, z=0, 



the total number of points in the group being 



2 3 + 4.2 2 + 6.2 + 4, i.e. ^^) 

 and so in general for n such equations the number of possible 



on l 



points of intersection will be — - — • 



As regards the resultant, we have 

 3 1 1 



x 1 y 2 z* 



x 4 y 4 z 4 



Hence the resultant of U, V, W, 12 is the same as that of 



V,Y,W, dt.^,f,z*,t*), 

 divided by the resultant of 



